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Appendix A: Tensors
Fig. A.1 Rotation of a
coordinate system (x, y, z)
by the Euler angles
(φ, θ, ψ) into the system
(X, Y, Z )
Y
Z
X
z
x
y
The matrix for the general rotation by the Euler angles is
⎛
⎝
cos ψ cos θ cos φ − sin ψ sin φ − sin φ cos θ cos ψ − cos φ sin ψ sin θ cos ψ
cos φ cos θ sin ψ + sin φ cos ψ − sin ψ cos θ sin φ + cos ψ cos φ sin θ sin ψ
− cos φ sin θ
sin φ sin θ
cos θ
⎞
⎠ .
(A.5)
A.3 Rank-n Tensors
Rank-0 Tensors
A tensor of rank 0 is also called a scalar. For example, the length v
2
1 +v
2
2 +v
2
3 of the vector v = (v 1 , v 2 , v 3 )
is a scalar since it is invariant under rotation of the coordinate system. However, ‘scalar’ is not equivalent
to ‘number’ since, e.g. the number v
2
1 + v
2
2 is not rotationally invariant.
Rank-1 Tensors
A tensor of rank 1 is a vector. It transforms under coordinate rotation D as
v
i = D i j v j .
(A.6)
Rank-2 Tensors
A tensor of rank 2 is also called a dyade and is a 3 × 3 matrix T that transforms under coordinate
rotation as
T
i j = D ik D jl T kl .
(A.7)
The physical meaning is the following: Two vectors s and r shall be related to each other via s i = T i j r j .
This could be, e.g., the current j and the electric field E that are connected via the tensor of conductivity
σ , i.e. j i = σ i j E j .